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Useful applications of the pistol
Snipers can use the pistol on a turn that they move and can be more accurate than a rifle at close range
Shotgunners can be more accurate at range
Characters that are out of ammo can keep firing instead of reloading on a critical turn
When pistol damage is enough to finish off an enemy, you can conserve ammo in main weapon ...

They make more sense in the remake (Lawton some good points). But when dealing with superhuman difficulty, every bit helps - even in the original.
In the original, they were tricky due to their lack of range and power, but still very useful for your grenadiers - using a rifle reduced your accuracy with grenades, and having a grenade in the other hand ...

If I just want to get done with missions quickly, I would use all shivs. In late game, I want to collect tons of money and finish all research. Unfortunately, occasionally, I got "missions". Otherwise I'll just get money after money after money.
So what?
I can finish missions much faster.
I can send alloy SHIVS or hovering shivs to rush enemies. No room ...